The generalized quasi-variational inequality problem with applications (Q1176165)

A variational inequality problem is proposed generalizing the problems considered by \textit{D. Chan} and \textit{J. S. Pang} [Math. Oper. Res. 7, 211--222 (1982; Zbl 0502.90080)] and \textit{J. Parida} and \textit{A. Sen} [J. Math Anal. Appl. 124, 73--81 (1987; Zbl 0615.49003)]. Let \(K\subseteq R^n\), \(C\subseteq R^m\). Given two maps \(\theta: K\times C\to R^n\) and \(\tau: K\times K\to R^n\) and a point-to-set mapping \(F: K\to C\), the problem is to find vectors \(\bar x\in X(\bar x)\), \(\bar y\in F(\bar x)\) such that \(\langle\theta(\bar x,\bar y),\tau(x,\bar x)\rangle\geq 0\) for all \(x\in X(\bar x)\). Existence theorems incorporating the cases of bounded and unbounded \(K\) are proved using the fixed point theorem by \textit{S. Eilenberg} and \textit{D. Montgomery} [Am. J. Math. 68, 214--222 (1946; Zbl 0060.40203)]. Applications to nonconvex mathematical programming and equilibrium programming problems are given.